A322389 -id:A322389 - cp's OEIS frontend

A259862 Triangle read by rows: T(n,k) = number of unlabeled graphs with n nodes and connectivity exactly k (n>=1, 0<=k<=n-1).

1, 1, 1, 2, 1, 1, 5, 3, 2, 1, 13, 11, 7, 2, 1, 44, 56, 39, 13, 3, 1, 191, 385, 332, 111, 21, 3, 1, 1229, 3994, 4735, 2004, 345, 34, 4, 1, 13588, 67014, 113176, 66410, 13429, 992, 54, 4, 1, 288597, 1973029, 4629463, 3902344, 1109105, 99419, 3124, 81, 5, 1, 12297299, 105731474, 327695586, 388624106, 162318088, 21500415, 820956, 9813, 121, 5, 1

Offset: 1

N. J. A. Sloane, Jul 08 2015

These are vertex-connectivities. Spanning edge-connectivity is A263296. Non-spanning edge-connectivity is A327236. Cut-connectivity is A327127. - Gus Wiseman, Sep 03 2019

			Triangle begins:
       1;
       1,       1;
       2,       1,       1;
       5,       3,       2,       1;
      13,      11,       7,       2,       1;
      44,      56,      39,      13,       3,     1;
     191,     385,     332,     111,      21,     3,    1;
    1229,    3994,    4735,    2004,     345,    34,    4,  1;
   13588,   67014,  113176,   66410,   13429,   992,   54,  4, 1;
  288597, 1973029, 4629463, 3902344, 1109105, 99419, 3124, 81, 5, 1;
  12297299,105731474,327695586,388624106,162318088,21500415,820956,9813,121,5,1;
  ...

Georg Grasegger, Table of n, a(n) for n = 1..78
Brendan McKay, confusion over k-connected graphs, posting to Sequence Fans Mailing List, Jul 08 2015.
Jens M. Schmidt, Combinatorial Data
Gus Wiseman, The graphs counted by row n = 5 (isolated vertices not shown).

Columns k=0..10 (up to initial nonzero terms) are A000719, A052442, A052443, A052444, A052445, A324234, A324235, A324088, A324089, A324090, A324091.
Row sums are A000088.
Number of graphs with connectivity at least k for k=1..10 are A001349, A002218, A006290, A086216, A086217, A324240, A324092, A324093, A324094, A324095.
The labeled version is A327334.
Cf. A002494, A263296, A322389, A326786, A327127, A327236.

A326786 Cut-connectivity of the set-system with BII-number n.

Original entry on oeis.org

0, 1, 1, 0, 2, 2, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 1, 1, 1, 1, 2, 2, 0, 0, 1, 1, 1, 1, 2, 0, 2, 0, 1, 1, 1, 1, 2, 0, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 0

Gus Wiseman, Jul 25 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18.

Elements of a set-system are sometimes called edges. The cut-connectivity of a set-system is the minimum number of vertices that must be removed (together with any resulting empty or duplicate edges) to obtain a disconnected or empty set-system. Except for cointersecting set-systems (A326853), this is the same as vertex-connectivity (A327051).

			Positions of first appearances of each integer, together with the corresponding set-systems, are:
     0: {}
     1: {{1}}
     4: {{1,2}}
    52: {{1,2},{1,3},{2,3}}
  2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}

Cf. A000120, A013922, A029931, A048793, A070939, A305078, A322388, A322389 (same for MM-numbers), A322390, A326031, A326701, A326749, A326753, A326787 (edge-connectivity), A327051 (vertex-connectivity).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
vertConn[y_]:=If[Length[csm[bpe/@y]]!=1,0,Min@@Length/@Select[Subsets[Union@@bpe/@y],Function[del,Length[csm[DeleteCases[DeleteCases[bpe/@y,Alternatives@@del,{2}],{}]]]!=1]]];
Table[vertConn[bpe[n]],{n,0,100}]

A327334 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and vertex-connectivity k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 4, 3, 1, 0, 26, 28, 9, 1, 0, 296, 490, 212, 25, 1, 0, 6064, 15336, 9600, 1692, 75, 1, 0, 230896, 851368, 789792, 210140, 14724, 231, 1, 0

Offset: 0

Gus Wiseman, Sep 01 2019

The vertex-connectivity of a graph is the minimum number of vertices that must be removed (along with any incident edges) to obtain a non-connected graph or singleton. Except for complete graphs, this is the same as cut-connectivity (A327125).

			Triangle begins:
    1
    1   0
    1   1   0
    4   3   1   0
   26  28   9   1   0
  296 490 212  25   1   0

Wikipedia, k-vertex-connected graph

The unlabeled version is A259862.
Row sums are A006125.
Column k = 0 is A054592, if we assume A054592(0) = A054592(1) = 1.
Column k = 1 is A327336.
Row sums without the first column are A001187, if we assume A001187(0) = A001187(1) = 0.
Row sums without the first two columns are A013922, if we assume A013922(1) = 0.
Cut-connectivity is A327125.
Spanning edge-connectivity is A327069.
Non-spanning edge-connectivity is A327148.
Cf. A322389, A327051, A327070, A327126, A327127, A327350.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
vertConnSys[vts_,eds_]:=Min@@Length/@Select[Subsets[vts],Function[del,Length[del]==Length[vts]-1||csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],vertConnSys[Range[n],#]==k&]],{n,0,5},{k,0,n}]

a(21)-a(35) from Robert Price, May 14 2021

A327125 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and cut-connectivity k.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 4, 3, 0, 1, 26, 28, 9, 0, 1, 296, 490, 212, 25, 0, 1, 6064, 15336, 9600, 1692, 75, 0, 1, 230896

Offset: 0

Gus Wiseman, Aug 25 2019

We define the cut-connectivity of a graph to be the minimum number of vertices that must be removed (along with any incident edges) to obtain a disconnected or empty graph, with the exception that a graph with one vertex and no edges has cut-connectivity 1. Except for complete graphs, this is the same as vertex-connectivity.

			Triangle begins:
    1
    0   1
    1   0   1
    4   3   0   1
   26  28   9   0   1
  296 490 212  25   0   1

After the first column, same as A327126.
The unlabeled version is A327127.
Row sums are A006125.
Column k = 0 is A054592, if we assume A054592(0) = 1.
Column k = 1 is A327114, if we assume A327114(1) = 1.
Row sums without the first column are A001187.
Row sums without the first two columns are A013922.
Different from A327069.
Cf. A259862, A322389, A326786, A327082, A327098, A327100, A327101.

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
cutConnSys[vts_,eds_]:=If[Length[vts]==1,1,Min@@Length/@Select[Subsets[vts],Function[del,csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],cutConnSys[Range[n],#]==k&]],{n,0,4},{k,0,n}]

a(21)-a(28) from Robert Price, May 20 2021

a(1) and a(2) corrected by Robert Price, May 20 2021

A327126 Triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with cut-connectivity k.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 3, 0, 1, 3, 28, 9, 0, 1, 40, 490, 212, 25, 0, 1, 745, 15336, 9600, 1692, 75, 0, 1

Offset: 0

Gus Wiseman, Aug 25 2019

We define the cut-connectivity of a graph to be the minimum number of vertices that must be removed (along with any incident edges) to obtain a disconnected or empty graph, with the exception that a graph with one vertex and no edges has cut-connectivity 1. Except for complete graphs, this is the same as vertex-connectivity.

			Triangle begins:
   1
   0   0
   0   0   1
   0   3   0   1
   3  28   9   0   1
  40 490 212  25   0   1

After the first column, same as A327125.
Column k = 0 is A327070.
Column k = 1 is A327114.
Row sums are A006129.
Different from A327069.
Row sums without the first column are A001187, if we assume A001187(0) = A001187(1) = 0.
Row sums without the first two columns are A013922.
Cf. A006125, A259862, A322389, A326786, A327114, A327127, A327198, A327237.

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
cutConnSys[vts_,eds_]:=If[Length[vts]==1,1,Min@@Length/@Select[Subsets[vts],Function[del,csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&cutConnSys[Range[n],#]==k&]],{n,0,4},{k,0,n}]

a(21)-a(27) from Robert Price, May 20 2021

A327051 Vertex-connectivity of the set-system with BII-number n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Gus Wiseman, Sep 02 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

The vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any empty or duplicate edges) to obtain a non-connected set-system or singleton. Except for cointersecting set-systems (A326853), this is the same as cut-connectivity (A326786).

			Positions of first appearances of each integer, together with the corresponding set-systems, are:
     0: {}
     4: {{1,2}}
    52: {{1,2},{1,3},{2,3}}
  2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}

Wikipedia, k-vertex-connected graph

Cut-connectivity is A326786.
Spanning edge-connectivity is A327144.
Non-spanning edge-connectivity is A326787.
The enumeration of labeled graphs by vertex-connectivity is A327334.
Cf. A000120, A013922, A029931, A048793, A070939, A259862, A322389, A323818, A326031, A327125, A327198, A327336.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
vertConnSys[vts_,eds_]:=Min@@Length/@Select[Subsets[vts],Function[del,Length[del]==Length[vts]-1||csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]]
Table[vertConnSys[Union@@bpe/@bpe[n],bpe/@bpe[n]],{n,0,100}]

A327098 BII-numbers of set-systems with cut-connectivity 1.

Original entry on oeis.org

1, 2, 8, 20, 21, 22, 23, 28, 29, 30, 31, 36, 37, 38, 39, 44, 45, 46, 47, 48, 49, 50, 51, 56, 57, 58, 59, 128, 260, 261, 262, 263, 272, 273, 276, 277, 278, 279, 280, 281, 284, 285, 286, 287, 292, 293, 294, 295, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309

Offset: 1

Gus Wiseman, Aug 21 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

We define the cut-connectivity (A326786, A327237), of a set-system to be the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a disconnected or empty set-system, with the exception that a set-system with one vertex and no edges has cut-connectivity 1. Except for cointersecting set-systems (A326853, A327039), this is the same as vertex-connectivity (A327334, A327051).

			The sequence of all set-systems with cut-connectivity 1 together with their BII-numbers begins:
   1: {{1}}
   2: {{2}}
   8: {{3}}
  20: {{1,2},{1,3}}
  21: {{1},{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  23: {{1},{2},{1,2},{1,3}}
  28: {{1,2},{3},{1,3}}
  29: {{1},{1,2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  31: {{1},{2},{1,2},{3},{1,3}}
  36: {{1,2},{2,3}}
  37: {{1},{1,2},{2,3}}
  38: {{2},{1,2},{2,3}}
  39: {{1},{2},{1,2},{2,3}}
  44: {{1,2},{3},{2,3}}
  45: {{1},{1,2},{3},{2,3}}
  46: {{2},{1,2},{3},{2,3}}
  47: {{1},{2},{1,2},{3},{2,3}}
  48: {{1,3},{2,3}}

A subset of A326749.
Positions of 1's in A326786.
BII-numbers for cut-connectivity 2 are A327082.
BII-numbers for non-spanning edge-connectivity 1 are A327099.
BII-numbers for spanning edge-connectivity 1 are A327111.
Integer partitions with cut-connectivity 1 are counted by A322390.
Labeled connected separable graphs are counted by A327114.
Connected separable set-systems are counted by A327197.
Cf. A000120, A048793, A070939, A322389, A326031, A327100, A327125.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
vertConnSys[sys_]:=If[Length[csm[sys]]!=1,0,Min@@Length/@Select[Subsets[Union@@sys],Function[del,Length[csm[DeleteCases[DeleteCases[sys,Alternatives@@del,{2}],{}]]]!=1]]];
Select[Range[0,100],vertConnSys[bpe/@bpe[#]]==1&]

A327114 Number of labeled simple graphs covering n vertices with cut-connectivity 1.

Original entry on oeis.org

0, 0, 0, 3, 28, 490, 15336, 851368, 85010976, 15615858960, 5388679220480, 3548130389657216, 4507988483733389568, 11145255551131555572992, 53964198507018134569758720, 514158235191699333805861463040

Offset: 0

Gus Wiseman, Aug 25 2019

nonn

The cut-connectivity of a graph is the minimum number of vertices that must be removed (along with any empty or duplicate edges) to obtain a disconnected or empty graph.

Andrew Howroyd, Table of n, a(n) for n = 0..50

Column k = 1 of A327126.
The unlabeled version is A052442, if we assume A052442(2) = 0.
Connected non-separable graphs are A013922.
BII-numbers for cut-connectivity 1 are A327098.
Set-systems with cut-connectivity 1 are counted by A327197.
Labeled simple graphs with vertex-connectivity 1 are A327336.
Cf. A001187, A054592, A259862, A322389, A322390, A326786, A327070, A327100, A327125.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
cutConnSys[vts_,eds_]:=If[Length[vts]==1,1,Min@@Length/@Select[Subsets[vts],Function[del,csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&cutConnSys[Range[n],#]==1&]],{n,0,3}]

seq(n)={my(g=log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n))); Vec(serlaplace(g-intformal(1+log(x/serreverse(x*deriv(g))))), -(n+1))} \\ Andrew Howroyd, Sep 11 2019

a(n) = A001187(n) - A013922(n), if we assume A001187(1) = 0.

A322387 Number of 2-vertex-connected integer partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 3, 1, 6, 2, 10, 8, 13, 9, 26, 14, 35, 28, 50, 37, 77, 54, 101, 84, 138, 110, 205, 149, 252, 222, 335, 287, 455, 375, 577, 522, 740, 657, 985

Offset: 1

Gus Wiseman, Dec 05 2018

An integer partition is 2-vertex-connected if the prime factorizations of the parts form a connected hypergraph that is still connected if any single prime number is divided out of all the parts (and any parts then equal to 1 are removed).

			The a(14) = 10 2-vertex-connected integer partitions:
  (14)  (8,6)   (6,4,4)   (6,3,3,2)  (6,2,2,2,2)
        (10,4)  (6,6,2)   (6,4,2,2)
        (12,2)  (10,2,2)

Wikipedia, k-vertex-connected graph

Cf. A013922, A095983, A218970, A275307, A304714, A304716, A305078, A305079, A322335, A322336, A322337, A322338, A322388, A322389, A322390.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
vertConn[y_]:=If[Length[csm[primeMS/@y]]!=1,0,Min@@Length/@Select[Subsets[Union@@primeMS/@y],Function[del,Length[csm[DeleteCases[DeleteCases[primeMS/@y,Alternatives@@del,{2}],{}]]]!=1]]];
Table[Length[Select[IntegerPartitions[n],vertConn[#]>1&]],{n,30}]

a(41)-a(42) from Jinyuan Wang, Jun 20 2020

A322390 Number of integer partitions of n with vertex-connectivity 1.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 8, 1, 7, 3, 11, 1, 14, 2, 18, 7, 21, 6, 35, 14, 43, 28, 65, 42, 96, 70, 141, 120, 205, 187, 315, 286, 445, 445, 657

Offset: 1

Gus Wiseman, Dec 05 2018

The vertex-connectivity of an integer partition is the minimum number of primes that must be divided out (and any parts then equal to 1 removed) so that the prime factorizations of the remaining parts form a disconnected (or empty) hypergraph.

			The a(14) = 7 integer partitions are (842), (8222), (77), (4442), (44222), (422222), (2222222).
The a(18) = 14 integer partitions:
  (9,9), (16,2),
  (8,8,2), (10,6,2),
  (8,4,4,2), (9,3,3,3),
  (4,4,4,4,2), (8,4,2,2,2),
  (3,3,3,3,3,3), (4,4,4,2,2,2), (8,2,2,2,2,2),
  (4,4,2,2,2,2,2),
  (4,2,2,2,2,2,2,2),
  (2,2,2,2,2,2,2,2,2).

Wikipedia, k-vertex-connected graph

Cf. A013922, A054921, A095983, A304714, A304716, A305078, A305079, A322335, A322338, A322387, A322389, A322391.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
vertConn[y_]:=If[Length[csm[primeMS/@y]]!=1,0,Min@@Length/@Select[Subsets[Union@@primeMS/@y],Function[del,Length[csm[DeleteCases[DeleteCases[primeMS/@y,Alternatives@@del,{2}],{}]]]!=1]]];
Table[Length[Select[IntegerPartitions[n],vertConn[#]==1&]],{n,20}]

cp's OEIS Frontend

A259862 Triangle read by rows: T(n,k) = number of unlabeled graphs with n nodes and connectivity exactly k (n>=1, 0<=k<=n-1).

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A326786 Cut-connectivity of the set-system with BII-number n.

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A327334 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and vertex-connectivity k.

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A327125 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and cut-connectivity k.

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A327126 Triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with cut-connectivity k.

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A327051 Vertex-connectivity of the set-system with BII-number n.

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A327098 BII-numbers of set-systems with cut-connectivity 1.

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A327114 Number of labeled simple graphs covering n vertices with cut-connectivity 1.

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Mathematica

PARI

Formula

A322387 Number of 2-vertex-connected integer partitions of n.